This question is fundamentally misinterpreting Godel's theorem. As far as we know, there is not a single "unprovable theorem", all theorems should be provable if they are meaningful, and provable in a natural system of mathematics, extending arithmetic with appropriate higher infinity axioms.
The objects constructed by Godel's theorem are unprovable in a given axiomatic system only, their natural function is as new axioms, which strengthen the axiom system in ways that allow them to be proven. Within set theory, you use axioms of higher infinity (inaccessible cardinals of various types) to do this.
Godel's theorem explains why you need these stronger axioms, and why these axioms produce new true theorems about arithmetic (although such enormously infinite objects are not particularly absolute themselves). But it does not mean that there is any question whatsoever that is undecidable in the sense of this question. In particular, the idea that P!=NP could be "undecidable" is not reasonable.
But P!=NP is not a simple statement, it says that there does not exist a computer program such that for any instance of 3SAT, it halts, and the eventual output satisfies this instance three sat. The structure of this statement is (not)(exists)(forall)(exists), or (forall)(exists)(forall), and this is Pi-3 (meaning three alternating quantifiers, starting with forall), and any true Pi-3 statement is hard to prove.
[ EDIT: Kaveh points out below that the innermost quantification is fake--- you don't need to do a quantification to check when a program in P halts--- it is guaranteed to halt in polynomial time. So if you phrase it properly, including the order n of the polynomial running time in the outermost "there exists", the innermost quantification is bounded above, and so it doesn't count as a step up. The proper statement: For all programs R and integers n such that R(p) stops changing in time less than $p^n$, there exists an instance of SAT indexed by integer P, such that the output of R(P) after Pn steps is not a solution to P. The proper phrasing shows that this is a $\Pi_2$ statement, not $\Pi_3$. $\Pi_2$ statements are not as difficult.]
The statement that every theorem should be provable from an axiom of higher infinity is not a theorem--- it is difficult to even formulate the statement precisely, because it it refers to a process of higher-infinity which cannot be formalized completely within an axiom system. But this is an article of mathematical faith--- articulated by Cohen in "Set Theory and the Continuum Hyptothesis". You might want to restrict this article of faith to Pi-1 statements, to statment
For a better discussion, see this: http://mathoverflow.net/questions/72062/what-are-some-proofs-of-godels-theorem-which-are-essentially-different-from-th/72151#72151